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交叉验证函数中的局部极小值AI 解读

SUMMARY The method of least squares cross-validation for choosing the bandwidth of a kernel density estimator has been the object of considerable research, through both theoretical analysis and simulation studies. The method involves the minimization of a certain function of the bandwidth. One of the less attractive features of this method, which has been observed in simulation studies but has not previously been understood theoretically, is that rather often the…

具有时间序列误差的核回归估计AI 解读

SUMMARY Suppose that data y 1, ..., yn are observed according to the model yi=f((i−12)/n)+ϵi,i=1,…,n, where f is a smooth function and {ϵ1,…,ϵn} is a sample from a zero-mean, covariance stationary time series. The problem of objectively choosing the bandwidth of a kernel estimate for f is addressed. It is shown both theoretically and by simulation that cross-validation produces extremely rough kernel estimates when the data are sufficiently positively correlated. This makes…

非参数回归中的稳定化带宽选择方法AI 解读

The problem of bandwidth selection for nonparametric kernel regression is considered. It is well recognized that the classical bandwidth selectors are subject to large sample variation. Due to the large variation, these selectors might not be very useful in practice. Based on the frequency-domain representation of the residual sum of squares (RSS), the source of the variation is pointed out. The observation leads to consideration of a procedure which stabilizes the RSS by…

样条函数与随机滤波AI 解读

Some relationships have been established between unbiased linear predictors of processes, in signal and noise models, minimizing the predictive mean square error and some smoothing spline functions. We construct a new family of multidimensional splines adapted to the prediction of locally homogeneous random fields, whose "$m$-spectral measure" (to be defined) is absolutely continuous with respect to Lebesgue measure and satisfies some minor assumptions. By considering…

利用指数族序列逼近概率密度函数AI 解读

Probability density functions are estimated by the method of maximum likelihood in sequences of regular exponential families. This method is also familiar as entropy maximization subject to empirical constraints. The approximating families of log-densities that we consider are polynomials, splines and trigonometric series. Bounds on the relative entropy (Kullback-Leibler distance) between the true density and the estimator are obtained and rates of convergence are…

非参数反卷积问题的最优收敛速度AI 解读

Deconvolution problems arise in a variety of situations in statistics. An interesting problem is to estimate the density $f$ of a random variable $X$ based on $n$ i.i.d. observations from $Y = X + \varepsilon$, where $\varepsilon$ is a measurement error with a known distribution. In this paper, the effect of errors in variables of nonparametric deconvolution is examined. Insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error…

回归模型整体与局部拟合诊断AI 解读

We concern ourselves with diagnostics for checking the overall and local goodness of fit of a model s(x) used in the regression of Y on x ∈ U = [0, 1] d . The model for s(x) is a functional form that depends on a finite number of unknown parameters. Two statistics, Λ and Λ w , are proposed that measure the level of agreement between the model fit to the data and the nonparametric kernel estimator on m preselected points in U. Conditions are given under which Λ and λ w , are…

消除核型非参数回归边界效应的几何方法AI 解读

We introduce a simple geometric method for removing edge effects from kernel-type nonparametric regression estimators. It involves reflecting the data set in two estimated points, thereby generating a new data set with three times the range of the original data. The usual kernel-type estimator may be applied to the new, enlarged data set, without any danger of edge effects. This technique is applicable generally to both regularly spaced and randomly spaced designs and admits…

非参数回归中平滑程度范围的选择AI 解读

Most nonparametric regression estimates smooth the observed data in the process of generating a suitable estimate. The resulting curve estimate is sensitive to the amount of smoothing introduced, and it is important to examine the estimate over a range of smoothing choices. An objective way to specify such a range is to estimate the optimal value of the smoothing parameter (bandwidth) with respect to some loss function and then report a confidence interval for this parameter…

灵活快速的自动平滑方法AI 解读

The choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators. The optimal choice depends on the data at hand. A data-driven bandwidth selection, close to the optimal one, would make these curve estimators objective, more reliable, and easier to use. Minimizing residual mean squared error criteria, such as cross-validation, have been frequently proposed to estimate the optimal smoothing parameter.…

利用傅里叶方法进行离散拉普拉斯平滑AI 解读

An approach to multidimensional smoothing is introduced that is based on a penalized likelihood with a modified discretized Laplacian penalty term. The choice of penalty simplifies computational difficulties associated with standard multidimensional Laplacian smoothing methods yet without compromising mean squared error characteristics, at least on the interior of the region of interest. For linear smoothing in hyper-rectangular domains, which has wide application in image…

样条平滑中交叉验证平滑参数的影响诊断AI 解读

This article addresses the problem of influence in estimating the smoothing parameter when fitting a univariate smoothing spline. Diagnostics are presented that can identify observations that locally influence the choice of smoothing parameter by generalized cross-validation, either through their case weights or observed responses. Generalized cross-validation and the diagnostic methods are given a frequency interpretation and illustrated with examples.

大量观测下核平滑条件分位数的计算AI 解读

Quantiles of a variable Y conditional on another variable X, when plotted against X, can be a useful descriptive tool. These plots give a quick impression of the functional form of the relation between X and the location, spread, and shape of the conditional distribution of Y. If several Y are observed for each X, then sample quantiles could be calculated for each X. The resulting quantile plot may be quite noisy, however, and smoothing across X may be desired. This article…

长度偏倚数据的核密度估计AI 解读

A new kernel density estimator for length biased data which derives from smoothing the nonparametric maximum likelihood estimator is proposed and investigated. It has various advantages over an alternative method suggested by Bhattacharyya, Franklin & Richardson (1988): it is necessarily a probability density, it is particularly better behaved near zero, it has better asymptotic mean integrated squared error properties and it is more readily extendable to related problems…

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Logistic 测量误差模型的半参数估计AI 解读

SUMMARY We describe semiparametric estimation and inference in a logistic regression model with measurement error in the predictors. The particular measurement error model consists of a primary data set in which only the response Y and a fallible surrogate W of the true predictor X are observed, plus a smaller validation data set for which (Y, X, W) are observed. Except for the underlying assumption of a logistic model in the true predictor, no parametric distributional…

核密度估计的一种可靠数据驱动带宽选择方法AI 解读

SUMMARY We present a new method for data-based selection of the bandwidth in kernel density estimation which has excellent properties. It improves on a recent procedure of Park and Marron (which itself is a good method) in various ways. First, the new method has superior theoretical performance; second, it also has a computational advantage; third, the new method has reliably good performance for smooth densities in simulations, performance that is second to none in the…

Bickel–Rosenblatt 拟合优度检验的功效与最优核AI 解读

Bickel and Rosenblatt proposed a procedure for testing the goodness of fit of a specified density to observed data. The test statistic is based on the distance between the kernel density estimate and the hypothesized density, and it depends on a kernel $K$, a bandwidth $b_n$ and an arbitrary weight function $a$. We study the behavior of the asymptotic power of the test and show that a uniform kernel maximizes the power when $a > 0$.

样条估计与其等价核估计的比较AI 解读

It has been observed that to a smoothing spline operator there corresponds an equivalent kernel operator; these two operators have been compared in a variety of norms [Cox (1984), Silverman (1984)]. In this paper, we refine the existing bounds for the particular case of the spline estimator considered in Rice and Rosenblatt (1983) and its corresponding equivalent kernel estimator. We obtain detailed asymptotic expressions for the bias and covariance functions of the two…

非参数回归的 Bootstrap 同时误差带AI 解读

Simultaneous error bars are constructed for nonparametric kernel estimates of regression functions. The method is based on the bootstrap, where resampling is done from a suitably estimated residual distribution. The error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints. Applications to an economic problem are given and comparison to both pointwise and Bonferroni-type bars is presented through a simulation study.

光滑单调回归函数的估计AI 解读

The problem of estimating a smooth monotone regression function $m$ will be studied. We will consider the estimator $m_{SI}$ consisting of a smoothing step (application of a kernel estimator based on a kernel $K$) and of a isotonisation step (application of the pool adjacent violator algorithm). The estimator $m_{SI}$ will be compared with the estimator $m_{IS}$ where these two steps are interchanged. A higher order stochastic expansion of these estimators will be given…

变量最优变换估计的收敛速度AI 解读

We consider here spline estimates of the optimal transformations of variables for multiple correlation and regression as dealt with in a recent paper by Breiman and Friedman. We show that we can construct estimates of the optimal transformations which have the same optimal rate of convergence as in the usual nonparametric estimation of a univariate function.

密度估计中的数据变换AI 解读

For the density estimation problem the global window width kernel density estimator does not perform well when the underlying density has features that require different amounts of smoothing at different locations. In this article we propose to transform the data with the intention that a global window width is more appropriate for the density of the transformed data. The density estimate of the original data is the “back-transform” by change of variables of the global…

离散化误差对核密度估计带宽选择的影响AI 解读

It is well known that the cross-validation score function of discretized data often tends to -∞ as the bandwidth tends to zero. This disturbing property causes some difficulty in applying cross-validation to discretized data. Since all data are rounded to some degree, the phenomenon limits the application of cross-validation in practice. Based on characteristic functions, the cause of the difficulty is explained, and a simple modification is suggested. Under some conditions,…